3.13.63 \(\int \sqrt {b d+2 c d x} (a+b x+c x^2) \, dx\) [1263]

Optimal. Leaf size=55 \[ -\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d}+\frac {(b d+2 c d x)^{7/2}}{28 c^2 d^3} \]

[Out]

-1/12*(-4*a*c+b^2)*(2*c*d*x+b*d)^(3/2)/c^2/d+1/28*(2*c*d*x+b*d)^(7/2)/c^2/d^3

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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \begin {gather*} \frac {(b d+2 c d x)^{7/2}}{28 c^2 d^3}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2),x]

[Out]

-1/12*((b^2 - 4*a*c)*(b*d + 2*c*d*x)^(3/2))/(c^2*d) + (b*d + 2*c*d*x)^(7/2)/(28*c^2*d^3)

Rule 697

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {align*} \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right ) \, dx &=\int \left (\frac {\left (-b^2+4 a c\right ) \sqrt {b d+2 c d x}}{4 c}+\frac {(b d+2 c d x)^{5/2}}{4 c d^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{12 c^2 d}+\frac {(b d+2 c d x)^{7/2}}{28 c^2 d^3}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.76 \begin {gather*} \frac {(d (b+2 c x))^{3/2} \left (-7 b^2+28 a c+3 (b+2 c x)^2\right )}{84 c^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*d + 2*c*d*x]*(a + b*x + c*x^2),x]

[Out]

((d*(b + 2*c*x))^(3/2)*(-7*b^2 + 28*a*c + 3*(b + 2*c*x)^2))/(84*c^2*d)

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Maple [A]
time = 0.49, size = 52, normalized size = 0.95

method	result	size
gosper	\(\frac {\left (2 c x +b \right ) \left (3 c^{2} x^{2}+3 b c x +7 a c -b^{2}\right ) \sqrt {2 c d x +b d}}{21 c^{2}}\)	\(46\)
derivativedivides	\(\frac {\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}}{4 c^{2} d^{3}}\)	\(52\)
default	\(\frac {\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}}{4 c^{2} d^{3}}\)	\(52\)
trager	\(\frac {\left (6 c^{3} x^{3}+9 b \,c^{2} x^{2}+14 a \,c^{2} x +b^{2} c x +7 a b c -b^{3}\right ) \sqrt {2 c d x +b d}}{21 c^{2}}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/4/c^2/d^3*(1/7*(2*c*d*x+b*d)^(7/2)+1/3*(4*a*c*d^2-b^2*d^2)*(2*c*d*x+b*d)^(3/2))

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Maxima [A]
time = 0.27, size = 46, normalized size = 0.84 \begin {gather*} -\frac {7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 3 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}}{84 \, c^{2} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

-1/84*(7*(2*c*d*x + b*d)^(3/2)*(b^2 - 4*a*c)*d^2 - 3*(2*c*d*x + b*d)^(7/2))/(c^2*d^3)

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Fricas [A]
time = 2.65, size = 58, normalized size = 1.05 \begin {gather*} \frac {{\left (6 \, c^{3} x^{3} + 9 \, b c^{2} x^{2} - b^{3} + 7 \, a b c + {\left (b^{2} c + 14 \, a c^{2}\right )} x\right )} \sqrt {2 \, c d x + b d}}{21 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/21*(6*c^3*x^3 + 9*b*c^2*x^2 - b^3 + 7*a*b*c + (b^2*c + 14*a*c^2)*x)*sqrt(2*c*d*x + b*d)/c^2

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Sympy [A]
time = 1.19, size = 48, normalized size = 0.87 \begin {gather*} \frac {\frac {\left (4 a c - b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {3}{2}}}{12 c} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{28 c d^{2}}}{c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)**(1/2)*(c*x**2+b*x+a),x)

[Out]

((4*a*c - b**2)*(b*d + 2*c*d*x)**(3/2)/(12*c) + (b*d + 2*c*d*x)**(7/2)/(28*c*d**2))/(c*d)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (47) = 94\).
time = 1.44, size = 228, normalized size = 4.15 \begin {gather*} \frac {420 \, \sqrt {2 \, c d x + b d} a b - \frac {140 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} a}{d} - \frac {70 \, {\left (3 \, \sqrt {2 \, c d x + b d} b d - {\left (2 \, c d x + b d\right )}^{\frac {3}{2}}\right )} b^{2}}{c d} + \frac {21 \, {\left (15 \, \sqrt {2 \, c d x + b d} b^{2} d^{2} - 10 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b d + 3 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )} b}{c d^{2}} - \frac {3 \, {\left (35 \, \sqrt {2 \, c d x + b d} b^{3} d^{3} - 35 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} d^{2} + 21 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} b d - 5 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )}}{c d^{3}}}{420 \, c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/420*(420*sqrt(2*c*d*x + b*d)*a*b - 140*(3*sqrt(2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*a/d - 70*(3*sqrt(
2*c*d*x + b*d)*b*d - (2*c*d*x + b*d)^(3/2))*b^2/(c*d) + 21*(15*sqrt(2*c*d*x + b*d)*b^2*d^2 - 10*(2*c*d*x + b*d
)^(3/2)*b*d + 3*(2*c*d*x + b*d)^(5/2))*b/(c*d^2) - 3*(35*sqrt(2*c*d*x + b*d)*b^3*d^3 - 35*(2*c*d*x + b*d)^(3/2
)*b^2*d^2 + 21*(2*c*d*x + b*d)^(5/2)*b*d - 5*(2*c*d*x + b*d)^(7/2))/(c*d^3))/c

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Mupad [B]
time = 0.48, size = 39, normalized size = 0.71 \begin {gather*} \frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,\left (28\,a\,c+3\,{\left (b+2\,c\,x\right )}^2-7\,b^2\right )}{84\,c^2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*d + 2*c*d*x)^(1/2)*(a + b*x + c*x^2),x)

[Out]

((b*d + 2*c*d*x)^(3/2)*(28*a*c + 3*(b + 2*c*x)^2 - 7*b^2))/(84*c^2*d)

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